Here's my method. I only replace the problem pins. Many of the pins are not used and I hate to apply too much heat to the board with my cheap de-soldering tool. So I don't replace the entire header. Do I really need to de-solder all those unused pins? Can I get a uniform look without replacing the entire header? Let's see.
First I remove the individual housings, using a razor blade to cut it from its neighbor and lift it off the pin. I could cut the pins, like the OP did, to make it easier to remove the housing. I didn't do that here, but may try that next time. I don't recall having any problems lifting the housings off the pins.
Next, I de-solder the pins.
In order to get a clean uniform look, I use Panduit header replacement parts. They are easy to cut from the longer strips using a razor blade. Trim them up a bit so they don't interfere with their neighbors. A single Panduit replacement header goes a long way when you only use a few pins for each repair.
I use a spare Panduit IDC connector to hold the new pins in place for soldering.
Here's how it looked when I was done soldering the new pins in.
